# Program

Though many of the ideas in higher category theory find their origins in homotopy theory — for instance as expressed by Grothendieck’s “homotopy hypothesis” — the subject today interacts with a broad spectrum of areas of mathematical research. Unforeseen descent, or local-to-global formulas, for familiar objects can be articulated in terms of higher invertible morphisms. Compatible associative deformations of a sequence of maps of spaces, or derived schemes, can putatively be represented by higher categories, as Koszul duality for E_n-algebras suggests. Higher categories offer unforeseen characterizing universal properties for familiar constructions such as K-theory. Manifold theory is natively connected to higher category theory and adjunction data, a connection that is most famously articulated by the recently proven Cobordism Hypothesis.
In parallel, the idea of "categorification'' is playing an increasing role in algebraic geometry, representation theory, mathematical physics, and manifold theory, and higher categorical structures also appear in the very foundations of mathematics in the form of univalent foundations and homotopy type theory. A central mission of this semester will be to mitigate the exorbitantly high "cost of admission'' for mathematicians in other areas of research who aim to apply higher categorical technology and to create opportunities for potent collaborations between mathematicians from these different fields and experts from within higher category theory.
Bibliography
Program Photography

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

higher categories

categorification

infinity-categories

algebraic K-theory

Factorization homology

cobordism hypothesis

synthetic higher categories

Grothendieck-Teichmuller

infinity-operads

topological quantum field theory

Mackey functors

**Primary Mathematics Subject Classification**

19Fxx - $K$K-theory in number theory [See also 11R70, 11S70]

55Pxx - Homotopy theory {For simple homotopy type, see 57Q10}

55Uxx - Applied homological algebra and category theory in algebraic topology [See also 18Gxx]

57Txx - Homology and homotopy of topological groups and related structures

81Txx - Quantum field theory; related classical field theories [See also 70Sxx]

**Secondary Mathematics Subject Classification**No Secondary AMS MSC